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## G = Q8×Dic5order 160 = 25·5

### Direct product of Q8 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — Q8×Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — Q8×Dic5
 Lower central C5 — C10 — Q8×Dic5
 Upper central C1 — C22 — C2×Q8

Generators and relations for Q8×Dic5
G = < a,b,c,d | a4=c10=1, b2=a2, d2=c5, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 152 in 70 conjugacy classes, 51 normal (14 characteristic)
C1, C2 [×3], C4 [×6], C4 [×5], C22, C5, C2×C4 [×3], C2×C4 [×4], Q8 [×4], C10 [×3], C42 [×3], C4⋊C4 [×3], C2×Q8, Dic5 [×2], Dic5 [×3], C20 [×6], C2×C10, C4×Q8, C2×Dic5, C2×Dic5 [×3], C2×C20 [×3], C5×Q8 [×4], C4×Dic5 [×3], C4⋊Dic5 [×3], Q8×C10, Q8×Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], Q8 [×2], C23, D5, C22×C4, C2×Q8, C4○D4, Dic5 [×4], D10 [×3], C4×Q8, C2×Dic5 [×6], C22×D5, Q8×D5, Q82D5, C22×Dic5, Q8×Dic5

Smallest permutation representation of Q8×Dic5
Regular action on 160 points
Generators in S160
(1 48 28 34)(2 49 29 35)(3 50 30 36)(4 41 21 37)(5 42 22 38)(6 43 23 39)(7 44 24 40)(8 45 25 31)(9 46 26 32)(10 47 27 33)(11 131 151 142)(12 132 152 143)(13 133 153 144)(14 134 154 145)(15 135 155 146)(16 136 156 147)(17 137 157 148)(18 138 158 149)(19 139 159 150)(20 140 160 141)(51 71 65 82)(52 72 66 83)(53 73 67 84)(54 74 68 85)(55 75 69 86)(56 76 70 87)(57 77 61 88)(58 78 62 89)(59 79 63 90)(60 80 64 81)(91 122 102 111)(92 123 103 112)(93 124 104 113)(94 125 105 114)(95 126 106 115)(96 127 107 116)(97 128 108 117)(98 129 109 118)(99 130 110 119)(100 121 101 120)
(1 68 28 54)(2 69 29 55)(3 70 30 56)(4 61 21 57)(5 62 22 58)(6 63 23 59)(7 64 24 60)(8 65 25 51)(9 66 26 52)(10 67 27 53)(11 122 151 111)(12 123 152 112)(13 124 153 113)(14 125 154 114)(15 126 155 115)(16 127 156 116)(17 128 157 117)(18 129 158 118)(19 130 159 119)(20 121 160 120)(31 82 45 71)(32 83 46 72)(33 84 47 73)(34 85 48 74)(35 86 49 75)(36 87 50 76)(37 88 41 77)(38 89 42 78)(39 90 43 79)(40 81 44 80)(91 142 102 131)(92 143 103 132)(93 144 104 133)(94 145 105 134)(95 146 106 135)(96 147 107 136)(97 148 108 137)(98 149 109 138)(99 150 110 139)(100 141 101 140)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160)
(1 91 6 96)(2 100 7 95)(3 99 8 94)(4 98 9 93)(5 97 10 92)(11 90 16 85)(12 89 17 84)(13 88 18 83)(14 87 19 82)(15 86 20 81)(21 109 26 104)(22 108 27 103)(23 107 28 102)(24 106 29 101)(25 105 30 110)(31 114 36 119)(32 113 37 118)(33 112 38 117)(34 111 39 116)(35 120 40 115)(41 129 46 124)(42 128 47 123)(43 127 48 122)(44 126 49 121)(45 125 50 130)(51 134 56 139)(52 133 57 138)(53 132 58 137)(54 131 59 136)(55 140 60 135)(61 149 66 144)(62 148 67 143)(63 147 68 142)(64 146 69 141)(65 145 70 150)(71 154 76 159)(72 153 77 158)(73 152 78 157)(74 151 79 156)(75 160 80 155)

G:=sub<Sym(160)| (1,48,28,34)(2,49,29,35)(3,50,30,36)(4,41,21,37)(5,42,22,38)(6,43,23,39)(7,44,24,40)(8,45,25,31)(9,46,26,32)(10,47,27,33)(11,131,151,142)(12,132,152,143)(13,133,153,144)(14,134,154,145)(15,135,155,146)(16,136,156,147)(17,137,157,148)(18,138,158,149)(19,139,159,150)(20,140,160,141)(51,71,65,82)(52,72,66,83)(53,73,67,84)(54,74,68,85)(55,75,69,86)(56,76,70,87)(57,77,61,88)(58,78,62,89)(59,79,63,90)(60,80,64,81)(91,122,102,111)(92,123,103,112)(93,124,104,113)(94,125,105,114)(95,126,106,115)(96,127,107,116)(97,128,108,117)(98,129,109,118)(99,130,110,119)(100,121,101,120), (1,68,28,54)(2,69,29,55)(3,70,30,56)(4,61,21,57)(5,62,22,58)(6,63,23,59)(7,64,24,60)(8,65,25,51)(9,66,26,52)(10,67,27,53)(11,122,151,111)(12,123,152,112)(13,124,153,113)(14,125,154,114)(15,126,155,115)(16,127,156,116)(17,128,157,117)(18,129,158,118)(19,130,159,119)(20,121,160,120)(31,82,45,71)(32,83,46,72)(33,84,47,73)(34,85,48,74)(35,86,49,75)(36,87,50,76)(37,88,41,77)(38,89,42,78)(39,90,43,79)(40,81,44,80)(91,142,102,131)(92,143,103,132)(93,144,104,133)(94,145,105,134)(95,146,106,135)(96,147,107,136)(97,148,108,137)(98,149,109,138)(99,150,110,139)(100,141,101,140), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,91,6,96)(2,100,7,95)(3,99,8,94)(4,98,9,93)(5,97,10,92)(11,90,16,85)(12,89,17,84)(13,88,18,83)(14,87,19,82)(15,86,20,81)(21,109,26,104)(22,108,27,103)(23,107,28,102)(24,106,29,101)(25,105,30,110)(31,114,36,119)(32,113,37,118)(33,112,38,117)(34,111,39,116)(35,120,40,115)(41,129,46,124)(42,128,47,123)(43,127,48,122)(44,126,49,121)(45,125,50,130)(51,134,56,139)(52,133,57,138)(53,132,58,137)(54,131,59,136)(55,140,60,135)(61,149,66,144)(62,148,67,143)(63,147,68,142)(64,146,69,141)(65,145,70,150)(71,154,76,159)(72,153,77,158)(73,152,78,157)(74,151,79,156)(75,160,80,155)>;

G:=Group( (1,48,28,34)(2,49,29,35)(3,50,30,36)(4,41,21,37)(5,42,22,38)(6,43,23,39)(7,44,24,40)(8,45,25,31)(9,46,26,32)(10,47,27,33)(11,131,151,142)(12,132,152,143)(13,133,153,144)(14,134,154,145)(15,135,155,146)(16,136,156,147)(17,137,157,148)(18,138,158,149)(19,139,159,150)(20,140,160,141)(51,71,65,82)(52,72,66,83)(53,73,67,84)(54,74,68,85)(55,75,69,86)(56,76,70,87)(57,77,61,88)(58,78,62,89)(59,79,63,90)(60,80,64,81)(91,122,102,111)(92,123,103,112)(93,124,104,113)(94,125,105,114)(95,126,106,115)(96,127,107,116)(97,128,108,117)(98,129,109,118)(99,130,110,119)(100,121,101,120), (1,68,28,54)(2,69,29,55)(3,70,30,56)(4,61,21,57)(5,62,22,58)(6,63,23,59)(7,64,24,60)(8,65,25,51)(9,66,26,52)(10,67,27,53)(11,122,151,111)(12,123,152,112)(13,124,153,113)(14,125,154,114)(15,126,155,115)(16,127,156,116)(17,128,157,117)(18,129,158,118)(19,130,159,119)(20,121,160,120)(31,82,45,71)(32,83,46,72)(33,84,47,73)(34,85,48,74)(35,86,49,75)(36,87,50,76)(37,88,41,77)(38,89,42,78)(39,90,43,79)(40,81,44,80)(91,142,102,131)(92,143,103,132)(93,144,104,133)(94,145,105,134)(95,146,106,135)(96,147,107,136)(97,148,108,137)(98,149,109,138)(99,150,110,139)(100,141,101,140), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,91,6,96)(2,100,7,95)(3,99,8,94)(4,98,9,93)(5,97,10,92)(11,90,16,85)(12,89,17,84)(13,88,18,83)(14,87,19,82)(15,86,20,81)(21,109,26,104)(22,108,27,103)(23,107,28,102)(24,106,29,101)(25,105,30,110)(31,114,36,119)(32,113,37,118)(33,112,38,117)(34,111,39,116)(35,120,40,115)(41,129,46,124)(42,128,47,123)(43,127,48,122)(44,126,49,121)(45,125,50,130)(51,134,56,139)(52,133,57,138)(53,132,58,137)(54,131,59,136)(55,140,60,135)(61,149,66,144)(62,148,67,143)(63,147,68,142)(64,146,69,141)(65,145,70,150)(71,154,76,159)(72,153,77,158)(73,152,78,157)(74,151,79,156)(75,160,80,155) );

G=PermutationGroup([(1,48,28,34),(2,49,29,35),(3,50,30,36),(4,41,21,37),(5,42,22,38),(6,43,23,39),(7,44,24,40),(8,45,25,31),(9,46,26,32),(10,47,27,33),(11,131,151,142),(12,132,152,143),(13,133,153,144),(14,134,154,145),(15,135,155,146),(16,136,156,147),(17,137,157,148),(18,138,158,149),(19,139,159,150),(20,140,160,141),(51,71,65,82),(52,72,66,83),(53,73,67,84),(54,74,68,85),(55,75,69,86),(56,76,70,87),(57,77,61,88),(58,78,62,89),(59,79,63,90),(60,80,64,81),(91,122,102,111),(92,123,103,112),(93,124,104,113),(94,125,105,114),(95,126,106,115),(96,127,107,116),(97,128,108,117),(98,129,109,118),(99,130,110,119),(100,121,101,120)], [(1,68,28,54),(2,69,29,55),(3,70,30,56),(4,61,21,57),(5,62,22,58),(6,63,23,59),(7,64,24,60),(8,65,25,51),(9,66,26,52),(10,67,27,53),(11,122,151,111),(12,123,152,112),(13,124,153,113),(14,125,154,114),(15,126,155,115),(16,127,156,116),(17,128,157,117),(18,129,158,118),(19,130,159,119),(20,121,160,120),(31,82,45,71),(32,83,46,72),(33,84,47,73),(34,85,48,74),(35,86,49,75),(36,87,50,76),(37,88,41,77),(38,89,42,78),(39,90,43,79),(40,81,44,80),(91,142,102,131),(92,143,103,132),(93,144,104,133),(94,145,105,134),(95,146,106,135),(96,147,107,136),(97,148,108,137),(98,149,109,138),(99,150,110,139),(100,141,101,140)], [(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110),(111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160)], [(1,91,6,96),(2,100,7,95),(3,99,8,94),(4,98,9,93),(5,97,10,92),(11,90,16,85),(12,89,17,84),(13,88,18,83),(14,87,19,82),(15,86,20,81),(21,109,26,104),(22,108,27,103),(23,107,28,102),(24,106,29,101),(25,105,30,110),(31,114,36,119),(32,113,37,118),(33,112,38,117),(34,111,39,116),(35,120,40,115),(41,129,46,124),(42,128,47,123),(43,127,48,122),(44,126,49,121),(45,125,50,130),(51,134,56,139),(52,133,57,138),(53,132,58,137),(54,131,59,136),(55,140,60,135),(61,149,66,144),(62,148,67,143),(63,147,68,142),(64,146,69,141),(65,145,70,150),(71,154,76,159),(72,153,77,158),(73,152,78,157),(74,151,79,156),(75,160,80,155)])

40 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 4G 4H 4I 4J 4K ··· 4P 5A 5B 10A ··· 10F 20A ··· 20L order 1 2 2 2 4 ··· 4 4 4 4 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 1 1 2 ··· 2 5 5 5 5 10 ··· 10 2 2 2 ··· 2 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + - + + - - + image C1 C2 C2 C2 C4 Q8 D5 C4○D4 D10 Dic5 Q8×D5 Q8⋊2D5 kernel Q8×Dic5 C4×Dic5 C4⋊Dic5 Q8×C10 C5×Q8 Dic5 C2×Q8 C10 C2×C4 Q8 C2 C2 # reps 1 3 3 1 8 2 2 2 6 8 2 2

Matrix representation of Q8×Dic5 in GL5(𝔽41)

 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 40 0
,
 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 20 38 0 0 0 38 21
,
 40 0 0 0 0 0 0 1 0 0 0 40 6 0 0 0 0 0 1 0 0 0 0 0 1
,
 32 0 0 0 0 0 16 25 0 0 0 39 25 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,1,0],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,20,38,0,0,0,38,21],[40,0,0,0,0,0,0,40,0,0,0,1,6,0,0,0,0,0,1,0,0,0,0,0,1],[32,0,0,0,0,0,16,39,0,0,0,25,25,0,0,0,0,0,1,0,0,0,0,0,1] >;

Q8×Dic5 in GAP, Magma, Sage, TeX

Q_8\times {\rm Dic}_5
% in TeX

G:=Group("Q8xDic5");
// GroupNames label

G:=SmallGroup(160,166);
// by ID

G=gap.SmallGroup(160,166);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,103,188,86,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^10=1,b^2=a^2,d^2=c^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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